Towards Minimal Barcodes

Abstract

Abstract. In the setting of persistent homology computation, a useful tool is the persistence barcode representation in which pairs of birth and death times of homology classes are encoded in the form of intervals. Starting from a polyhedral complex K (an object subdivided into cells which are polytopes) and an initial order of the set of vertices, we are concerned with the general problem of searching for filters (an order of the rest of the cells) that provide a minimal barcode representation in the sense of having minimal number of "k-significant" intervals, which correspond to homology classes with life-times longer than a fixed number k. As a first step, in this paper we provide an algorithm for computing such a filter for k = 1 on the Hasse diagram of the poset of faces of K.

An entropy-based persistence barcode

“…While one would ideally want to balance the minimization with a penalty term of the number of non-significant intervals [18], further knowledge of statistical distribution of the long intervals is required and postponed to future work. This paper completes the work proposed in [12] by providing additional insights, examples, results and proofs. A new definition of "minimal barcode" is given using the notion of entropy, and an algorithm for computing persistence barcodes with small entropy starting from a given filtration is provided.…”

“…On the other hand, if someone examines in depth typical cases where persistent homology is used, one will be faced with the inherent problem of "noise" in the persistence barcodes. A similar problem was posed in [6,12] where various algorithmic results were presented. Those problems take the form of non-significant topological holes in sensor networks and inefficiencies of the filtration coming from the construction of the Rips (or Čech) complex in data sets.…”

“…Following this idea, a notion of k-significant intervals was introduced in [12]: Fix k 4 0, an interval ½s; tÞ is k-significant if k o t À s. Persistence barcodes associated to different filters of K can be compared by the amount of k-significant intervals they contain.…”

“…Definition 5 (Rocio Gonzalez-Diaz et al [12]). A persistence barcode associated with a filter F A F K is k-minimal if the persistence barcode associated with any other filter in F K contains greater or equal number of k-significant intervals.…”